Signals and Systems
Lecture 3: Discrete-Time LTI Systems: Frequency DomainConcepts
Dr. Guillaume Ducard
Fall 2018
based on materials from: Prof. Dr. Raffaello D’Andrea
Institute for Dynamic Systems and Control
ETH Zurich, Switzerland
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Outline
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionDiscrete-time frequency range
Outline
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
3 / 33
Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionDiscrete-time frequency range
Recall: a complex number z ∈ C can be written in
Polar form
z = |z|ejΩ,
where j2 = −1,|z| ≥ 0 is the magnitude of z,
and Ω ∈ (−π, π] is the phase of z. Note thatΩ can be defined in any 2π interval, such as[0, 2π).
Cartesian form
z = a+ jb,
where a, b ∈ R.
Re(z)
Im(z)
Ω0
z0
|z0|
a0
b0
The two forms are related inthat |z| =
√a2 + b2 and
Ω = atan2(b, a), whereatan2 is the four-quadrantarc-tangent function.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionDiscrete-time frequency range
Complex exponential sequence
Consider the complex exponential sequence with
x[n] = zn for all n,where z ∈ C. (1)
This can be rewritten using Euler’s formula, ejΩ = cosΩ + j sinΩ,as:
x[n] = zn
= (|z|ejΩ)n
= |z|nejΩn
= |z|n(cos(Ωn) + j sin(Ωn)),
where Ω is the discrete-time frequency of the sequence’s oscillatorycomponent.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionDiscrete-time frequency range
Outline
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionDiscrete-time frequency range
Discrete-time frequency range
CT sinusoids
CT sinusoids, for example cos(ωt), are distinct for all frequenciesω ≥ 0.
DT sinusoids
cos((Ω + 2πk)n) = cos(Ωn+ 2πkn) = cos(Ωn), for k ∈ Z.
Consequences
1 DT frequencies Ω and Ω+ 2πk give identical sequences for allintegers k.
2 DT frequency is therefore restricted to an interval oflength 2π.
3 Usually the intervals [0, 2π) or (−π, π] are chosen, howeverany interval of length 2π is appropriate.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionPeriodicity constraint
Outline
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionPeriodicity constraint
A sequence x is called periodic
if there exists an integer N > 0 such that x[n+N ] = x[n] for all n.
It follows that x[n+mN ] = x[n] for all m ∈ Z.
The smallest N for which x[n+N ] = x[n] for all n is calledthe fundamental period.
ExampleThe following signal is periodic with fundamental period 2.
n
x[n]
−1 0 1 2 39 / 33
Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionPeriodicity constraint
Outline
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionPeriodicity constraint
Periodicity constraint
A CT sinusoid cos(ωt) is always periodic with a period of T = 2π|ω| .
If this periodic CT signal is uniformly sampled with a samplingtime of Ts, the resulting DT signal x[n] = cos(Ωn), withfrequency Ω = ωTs, is periodic if and only if
Ω
2π=
m
N, for some integers m and N.
In other words, x[n] is periodic if and only if Ω/2π isrational. (You will prove this in the problem sets.)
Furthermore, if mN
is an irreducible fraction, then N is thefundamental period of the signal.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
Table of Content
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
Eigenfunctions of LTI Systems
Let the LTI system G have input u[n] = zn, with z ∈ C. Thenthe output sequence y[n] is calculated using the convolutiony = h ∗ u to be:
y[n] = Gzn =∞∑
k=−∞
h[k]zn−k =∞∑
k=−∞
h[k]znz−k
=
∞∑
k=−∞
h[k]z−kzn
= H(z)zn,
where
H(z) :=
∞∑
k=−∞
h[k]z−k.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
Eigenfunctions of LTI Systems
It follows that:
zn is an eigenfunction of G
and H(z) is the corresponding eigenvalue.
Recall the CT counterpart:
y = Gu, with u(t) = est gives y(t) = H(s)est, where H(s) is theLaplace Transform of the impulse response of G.⇒ In CT, the Laplace Transform enables the analysis of CTsystems in the frequency domain.
The equivalent DT transformation
is known as the z-transform and allows DT systems to be analyzedin the frequency domain. We investigate this transformation next.
14 / 33
Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Outline
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Given a sequence x[n], its z-transform X(z) is defined as
X(z) :=∞∑
n=−∞
x[n]z−n, z ∈ C.
The transform pair, used to indicate the relationship between thesequence x[n] and its z-transform X(z), is denoted asx[n] ←→ X(z).
Remark:This is a slight abuse of notation as we use X(z) to denote both thez-transform of x[n] and its value at a particular z; however, this notationhelps to distinguish between the z-transform and the DT Fourier transform(introduced in Lecture 4), and it should be clear from context whether thefunction or the function value is intended.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Properties
Given that a sequence x[n] is related to its z-transform X(z) asx[n] ←→ X(z), the following properties hold:
• Linearity: a1x1[n]+ a2x2[n] ←→ a1X1(z) + a2X2(z)
• Time-shifting x[n − 1] ←→ z−1X(z)
x[n + 1] ←→ zX(z)
• Convolution: x1[n] ∗ x2[n] ←→ X1(z) ·X2(z)
• Accumulation: n∑
k=−∞
x[k] ←→z
z − 1X(z)
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Outline
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Recall: Geometric series
Sn = a+ a · r + a · r2 + · · ·+ a · rn−1 =
n−1∑
k=0
a · rk = a
n−1∑
k=0
rk = a
1− rn
1− r,
where the first term is a ∈ R and the term r ∈ R is called common ratio.
Proof
Sn = a+ a · r + a · r2 + · · ·+ a · rn−1
r · Sn = a · r + a · r2 + a · r3 + · · ·+ a · rn
Sn − r · Sn = a− a · rn
Sn · (1− r) = a(1− rn)
Sn = a1− rn
1− rwith r 6= 1 .
If the common ratio has a module strictly smaller than one (|r| < 1), the seriesconverges to
limn→∞
n−1∑
k=0
a rk = lim
n→∞a1− rn
1− r= a
1
1− r. (2)
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Consider the sequence x[n] defined by :
x[n] =
an n ≥ 0, a ∈ R, a 6= 0
0 otherwise
Its z-transform X(z) is given as:
X(z) =
∞∑
n=0
anz−n =
∞∑
n=0
(a
z
)n
.
It can be shown that the above sum converges iff |a/z| < 1, or|z| > |a|.In that case,
X(z) =1
1− az
=z
z − a(3)
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Now, consider a different sequence
x[n] =
−an n < 0, a ∈ R, a 6= 0
0 otherwise
X(z) = −−1∑
n=−∞
anz−n = −∞∑
n=1
a−nzn = −∞∑
n=1
(z
a
)n
.
The above sum converges iff |z/a| < 1, or |z| < |a|. In that case:
X(z) =− z
a
1− za
=−z
a− z=
z
z − a
The z-transforms of the two sequences are identical!
The difference is the region of convergence (ROC), the values of z for whichX(z) converges.Therefore, strictly speaking, the z-transform must also include the ROC touniquely specify the sequence in the time domain.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Outline
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Transfer functions of LTI systems
For an LTI system with impulse response h[n] we have
y[n] = u[n] ∗ h[n] ←→ Y (z) = U(z)H(z)
∴H(z) =Y (z)
U(z).
H(z) is the transfer function of the system. It is the z-transform ofits impulse response.
1 Therefore, as previously discussed, we also need to know theROC of H(z) to uniquely capture the input-output behaviorof the system,
2 however, in practice this is not necessary (as shown later).
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Transfer function from LCCDE
Consider a system described by the LCCDE:
N∑
k=0
aky[n− k] =
M∑
k=0
bku[n− k]
Using the z-transform, we can write
N∑
k=0
aky[n− k] =M∑
k=0
bku[n− k]←→N∑
k=0
akz−kY (z) =
M∑
k=0
bkz−kU(z),
and can then write the system’s transfer function as
Y (z)
U(z)=
b0 + b1z−1 + . . . + bMz−M
a0 + a1z−1 + . . .+ aNz−N= H(z).
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Let us compare this with the shift operator z introduced in Lecture1:
N∑
k=0
akz−ky =
M∑
k=0
bkz−ku
y
u=
b0 + b1z−1 + . . . + bMz
−M
a0 + a1z−1 + . . .+ aNz−N
= H(z),
where H(z) can now be thought of as an operator acting onsequences. This is due to the relationship between z, whichoperates in the time domain and z, which operates in thefrequency domain:
x[n+ 1]←→ zX(z)
x[n+ 1] = (zx)[n].
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Transfer function from a state-space description
The transfer function H(z) can also be determined starting from astate-space description:
q[n+ 1] = Aq[n] +Bu[n]
y[n] = Cq[n] +Du[n]
l
zQ(z) = AQ(z) +BU(z)
→ Q(z) = (zI −A)−1BU(z)
Y (z) = CQ(z) +DU(z)
→ H(z) =Y (z)
U(z)= C(zI −A)−1B +D
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Outline
1 Complex Exponential SequencesDefinitionDiscrete-time frequency range
2 Periodic SignalsDefinitionPeriodicity constraint
3 Eigenfunctions of LTI Systems
4 The z-TransformDefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Stability and causality
We will address the subtle aspects of stability and causality with anexample. Consider the following difference equation:
y[n] = ay[n− 1] + u[n].
We can readily calculate its impulse response as
h[n] =
an for n ≥ 0
0 otherwise
h = . . . , 0, 0, 1↑, a, a2, a3, . . . .
We therefore have that the system captured by the above differenceequation is stable if |a| < 1, since then
∑∞n=−∞ |h[n]| <∞.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Stability and causality
Let us now rewrite the above difference equation in a different way:
y[n− 1] =1
ay[n]−
1
au[n].
If we now go backwards in time, we observe that the following isalso a valid impulse response:
h[n] =
−an for n < 0
0 otherwise
h = . . . ,−1
a3,−
1
a2,−
1
a, 0↑, 0, . . . .
So in this instance, the system is stable if |a| > 1. The difference isthat in this interpretation of the difference equation, the system isnot causal (the impulse response is non-zero for negative timeindices).
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Stability and causality
We have, in fact, seen this example before:
y[n] = ay[n− 1] + u[n]
←→ Y (z) = az−1Y (z) + U(z)
zY (z) = aY (z) + zU(z)
Y (z)
U(z)= H(z) =
z
z − a.
Recall that H(z) has two possible inverse z-transforms, onecausal and one non-causal, depending on its ROC.
In practice, we usually require the system to be causal, andthus it is not necessary to specify the ROC of H(z).
The conclusion is that stability and causality are intertwined.30 / 33
Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Therorem Given a transfer function H(z), there exists a stable and causalinterpretation for the underlying system iff all poles of H(z) are inside the unitcircle. That is, given pole p (a value p for which |H(p)| =∞), then |p| < 1.
Proof We will show this result for the special case that H(z) is a rationaltransfer function, as would arise from LCCDE and SS descriptions of a system.We also assume that the finite poles of H(z) are distinct. In that case, H(z)can be factored as:
H(z) =
Q∑
k=0
ckzk +
R∑
k=1
βk
z
z − pk.
We observe that in order for H(z) to have a causal interpretation, ck = 0 fork > 0. If this is not the case, y will always depend on future values of u; thiscorresponds to poles of H(z) at z =∞. Therefore, consider
H(z) = c0 +
R∑
k=1
βk
z
z − pk.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
We can write down a causal impulse response h using the above z-transformand the z-transform of the unit impulse sequence δ[n]←→ 1:
h[n] =
c0 +R∑
k=1
βk n = 0
R∑
k=1
βkpnk n > 0
0 otherwise
It can then be shown that
∞∑
n=0
|h[n]| <∞ iff |pk| < 1.
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Complex Exponential SequencesPeriodic Signals
Eigenfunctions of LTI SystemsThe z-Transform
DefinitionConvergence and non-uniquenessTransfer functions of LTI systemsStability and causality
Non-causal systems
Causality is, however, not always necessary. Consider
y[n] =1
3(u[n+ 1] + u[n] + u[n− 1]).
There is no causal interpretation of this LCCDE. But in fact, this is a simplemoving average filter that can be used to smooth signals in post-processingapplications. In other words, causality is not a requirement in non-real-timeapplications. We have the following result:
Theorem Given the transfer function H(z), there exists a stable, but notnecessarily causal, interpretation for the underlying system iff H(z) has nopoles on the unit circle. That is, if |H(p)| =∞, then |p| 6= 1.
Proof.
We now prove one direction of the above statement: If there is a stable interpretation for the system, then H(z)has no poles on the unit circle. Let |z| = 1. It follows that
|H(z)| =
∣
∣
∣
∣
∣
∣
∞∑
n=−∞
h[n]z−n
∣
∣
∣
∣
∣
∣
≤∞∑
n=−∞
|h[n]| < ∞.
As required, H(z) has no pole p with |p| = 1. The proof of the other direction, that if H(z) has no pole p with
|p| = 1 there exists a stable interpretation for the system, is similar to the proof of the previous theorem. You will
prove this in the problem sets. 33 / 33